Remark 7.8 As it can be viewed for instance in [1]}, the Morse–Smale transversality assumption allows to show that there exists a way for constructing a compactification of these moduli spaces of gradient flow lines. Without going into too much details, it can be done by applying a series of fibered products with respect to the endpoint maps \(u\)  and \(ł\) . Therefore, as consequence of Proposition REF , [1]} and [3]} we may extend the Lie groupoid structure previously described to the respective compactifications. Here it is important to have in mind that \(u\)  and \(l\)  define Lie groupoid morphisms where \(u\)  is composed by locally trivial fibrations.

